IJPAM: Volume 53, No. 2 (2009)

PRODUCT OF $\delta ^{(k-1)}(x).\delta ^{(l-1)}(x)$ AND $\delta ^{(k-1)}(x-a).\delta ^{(l-1)}(x-b) $

Manuel A. Aguirre
Núcleo Consolidado Matemática Pura y Aplicada
Facultad de Ciencias Exactas
Universidad Nacional del Centro
Tandil, Provincia de Buenos Aires, ARGENTINA
e-mail: maguirre@exa.unicen.edu.ar

Abstract.One of the problem in distribution theory is the lack of definitions for products and power of distributions in general. In physics (c.f. [#!Ga!#], p. 141), oneself finds the need to evaluate $ \delta ^{2} $ when calculating the transition rates of certain particle interactions. Chenkuan Li (see [#!L!#]) derives that $ \delta ^{2}(x)=0 $ on even-dimension space by applying the Laurent expansion of $ r=1 $. Koh and Li in [#!K!#] give a sense to distribution $ \delta ^{k} $ and $ (\delta ^{\lq })^{k} $ for some $ k $, using the concept of neutrix limit. Aguirre in [#!A!#], gives a sense to distributional product of $ \delta ^{(m)}(x).\delta ^{(l)}(x) $, using the Hankel transform of generalized function of $ \delta ^{(m)}(x) $. In this paper using the Fourier transform of $ \delta ^{(k-1)}(x) $ we obtain formulae for the distributional product of $\delta ^{(k-1)}(x).\delta ^{(l-1)}(x)$ and $ %
\delta ^{(k-1)}(x-a). \delta ^{(l-1)}(x-b), $ where $ k $ and $ l=1,2,3,. . $. As consequence we give a sense at the following product: $ \delta
^{2}(x),(\delta (x))^{k},(\delta ^{(k-1)}(x))^{m},(\delta (x-a))^{t} $ and $ %
(\delta ^{k}(x-a))^{m} $. Finally, we write formulae relations with distributional products of $ (\delta ^{k}(P(x_{1},. . . x_{n}))^{2} $ and $ %
(\delta ^{k}(m^{2}+P(x_{1},. . . x_{n}))^{2} $ where $ P(x_{1},. . . x_{n}) $ is defined by ([*]).

Received: March 31, 2009

AMS Subject Classification: 47Bxx, 45P05, 47G10, 32A25, 32M15

Key Words and Phrases: distribution theory, Laurent expansion, Hankel transform

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 53
Issue: 2